If the radius of a circle is doubled, what effect does this have on the area of the circle? On the circumference?  Explain why your result makes sense.

If the radius of a circle is doubled, the area of the circle will be quadrupled and the circumference will also be doubled. This makes sense because the area of a circle is directly proportional to the square of the radius of the circle (if the radius of a circle is increased x times, its area will be increased to x^2 times the original area), whereas the circumference is proportional to the size of the radius (if the radius of a circle is increased x times, its circumference will increase x times).

What you have to remember here is the formulas for area and circumference of a circle.

The area of a circle is equal to the square of its radius times pi. The circumference of a circle is equal to the diameter times pi.

So, let's look at what happens if you double the radius of a circle -- say from 2 to 4.

The area will go from 12.56 to 50.24. This means that it has quadrupled. The reason for this is that you square the radius. So when you double the radius, the area goes up by 4 times because 2 squared is 4. The area will always go up by the square of how much the radius goes up.

By contrast, the circumference will only double -- from 12.56 to 25.12 because you do not square the radius (or diameter) -- you just multiply it by pi. So the circumference will go up by the same percent that the radius does.

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